Mathematical Methods for Physics and Engineering : A Comprehensive Guide

REPRESENTATION THEORY

4 mm IQR,R′ mx,my md,md′

A 1 11 1 1 1

A 2 11 1 − 1 − 1

B 1 11 − 11 − 1

B 2 11 − 1 − 11

E 2 −20 0 0

Table 29.4 The character table deduced for the group 4mm. For an explana-

tion of the entries in bold see the text.

with the characters of A 1 requires that

1(1)(1) + 1(1)(1) + 2(1)(p) + 2(1)(q) + 2(1)(r)=0.

The only possibility is that two ofp,q,andrequal−1 and the other equals +1. This
can be achieved in three different ways, corresponding to the need to find three further
different one-dimensional irreps. Thus the first four lines of entries in character table 29.4
can be completed. The final line can be completed by requiring it to be orthogonal to the
other four. Property (v) has not been used here though it could have replaced part of the
argument given.

29.9 Group nomenclature

The nomenclature of published character tables, as we have said before, is erratic

and sometimes unfortunate; for example, oftenEis used to represent, not only

a two-dimensional irrep, but also the identity operation, where we have usedI.

Thus the symbolEmight appear in both the column and row headings of a

table, though with quite different meanings in the two cases. In this book we use

roman capitals to denote irreps.

One-dimensional irreps are regularly denoted by A and B, B being used if a

rotation about the principal axis of 2π/nhas character−1. Herenis the highest

integer such that a rotation of 2π/nis a symmetry operation of the system, and

the principal axis is the one about which this occurs. For the group of operations

on a square,n= 4, the axis is the perpendicular to the square and the rotation

in question isR. The names for the group, 4mmandC 4 v, derive from the fact

that herenis equal to 4. Similarly, for the operations on an equilateral triangle,

n= 3 and the group names are 3mandC 3 v, but because the rotation by 2π/3has

character +1 in all its one-dimensional irreps (see table 29.1), only A appears in

the irrep list.

Two-dimensional irreps are denoted by E, as we have already noted, and three-

dimensional irreps by T, although in many cases the symbols are modified by

primes and other alphabetic labels to denote variations in behaviour from one

irrep to another in respect of mirror reflections and parity inversions. In the study

of molecules, alternative names based on molecular angular momentum properties

are common. It is beyond the scope of this book to list all these variations, or to